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Attribution| 4.0 International LicenseA Control Delay Differential Equations Model of Evolution of Normal and Leukemic Cell Populations
Under Treatment
I. Rădulescu, Doina Cândea, Andrei Halanay
To cite this version:
I. Rădulescu, Doina Cândea, Andrei Halanay. A Control Delay Differential Equations Model of
Evolution of Normal and Leukemic Cell Populations Under Treatment. Christian Pötzsche; Clemens
Heuberger; Barbara Kaltenbacher; Franz Rendl. System Modeling and Optimization : 26th IFIP TC 7
Conference, CSMO 2013, Klagenfurt, Austria, September 9–13, 2013, Revised Selected Papers, AICT-
443, Springer Berlin Heidelberg, pp.257-266, 2014, IFIP Advances in Information and Communication
Technology, 978-3-662-45503-6. �10.1007/978-3-662-45504-3_25�. �hal-01286684�
A control delay differential equations model of evolution of normal and leukemic cell
populations under treatment
?I. Rodica R˘adulescu??, Doina Cˆandea, and Andrei Halanay POLITEHNICA University of Bucharest
Department of Mathematics and Informatics Splaiul Independentei 313 RO-060042 Bucharest, Romania
Abstract. The dynamics and evolution of leukemia is determined by the interactions between normal and leukemic cells populations at every phase of the development of hematopoietic cells. For both types of cell populations, two subpopulations are considered, namely the stem-like cell population (i.e. with unlimited self-renew ability) and a more ma- ture, differentiated one, possessing only the capability to undergo limited reproduction. Treatment effects are included in the model as functions of time and a cost functional is considered. The optimal control is obtained using a discretization scheme. Numerical results are discussed in relation to the medical interpretation.
Keywords: leukemia, asymmetric division, competition, optimal con- trol, treatment
AMS Classification: 34K35, 37N25, 92C50, 93C23.
1 Introduction
For the description of biological processes implied in hematopoiesis, a mathe- matical model that includes time delays will be used. It is based on the mass action principle, in the spirit of [1, 2, 4, 12, 16] and [20]. Other authors ([14, 15]) used the more simple model from [18] for the dynamics of hematopoietic stem cells (HSC).
Chronic Myelogenous Leukemia (CML), also known as Chronic Granulocytic Leukemia, is a cancer of white blood cells. It is a clonal marrow stem cells dis- order in which the main characteristic is the proliferation of granulocytes (neu- trophils, eosinophils and basophils) and of their precursors in the bone marrow and the accumulation of these cells in the blood. It is a type of myeloprolifera- tive disease associated to a chromosomal translocation called the Philadelphia chromosome (see also [6, 19]) presenting the oncogene BCR-ABL that encodes
?This work was supported by CNCS-ROMANIA Grant ID-PCE-2011-3-0198.
?? Corresponding author. E-mail:nicola rodica@yahoo.com
a tyrosine kinase protein. Tyrosine kinases are enzymes that play an important role in tumor development by supporting cell growth through phosphorylation of signaling proteins [11]. Understanding the molecular mechanism of CML per- mitted the development of specific tyrosine kinase inhibitors (TKIs) as imatinib (Gleevec), dasatinib or nilotinib. The standard first line therapy is nowadays imatinib, which acts through competitive inhibition at the ATP-binding site of the BCR-ABL enzyme, leading to the inhibition of tyrosine phosphorylation of proteins involved in BCR-ABL signal transduction [7]. The molecular effect of imatinib is mainly the inhibition of cell proliferation of BCR-ABL–positive cells but, there are experimental evidences that, in the mature cell lines, the inhibition of cell proliferation is followed by apoptosis [11]. Although imatinib has a very good successful rate, there are many experimental evidences attesting it does not affect quiescent stem cells deep in the bone marrow, and the consequence is the disease reapers after the treatment is stopped.
In this paper, we study an optimal control delay differential equation model of four cell populations, namely two healthy and two leukemic. For these classes of cells, we consider a population of mature cells which lost their self-renew ability and a population of stem-like cells involving a larger category consisting of proliferating stem and progenitor cells with self-renew capacity. The emphasis in this optimal control model is on establishing treatment strategies, considering the competition of healthy vs. CML cell populations and three types of division that a stem-like cell can exhibit: self-renew, asymmetric division and differentiation ([4, 17, 20, 21, 24]).
Of course, besides a correct mathematical model for the time evolution of the studied cell populations, it is very important to model the treatment effect as accurate as one can. Obviously, different drugs have different effects: some affect not only leukemic populations but also healthy ones (the cytotoxic ones), some kill the cells while others only delay or stop the division process (the cytostatic ones). We take into account here the standard treatment protocol with imatinib which, as we specified, acts by inhibiting the BCR-ABL signal transduction. In this way, it restrains the proliferative advantage of the CML cells and healthy cells regain their advantage. Moreover, it is straightforward to say that imatinib restores most of the abnormal functions of the CML cells, and the most important function affected by the drug is the division process of CML cell population. However, it is uncertain to what extent it affects all three kinds of division and therefore, we consider here the hypothesis that imatinib influences self-renew, asymmetric division and differentiation equally.
2 Description of the model
In the present paper, is assumed that the hematopoietic stem cells that are considered are in the proliferative phase or spend a short time into the resting phase. These cells are called, following [17], Short-Term Hematopoietic Stem Cells (ST-HSC). In what followsx1 denotes the density of short-term stem-like
A control DDE model of normal and leukemic cells under treatment 257
healthy cells ,x2the density of mature healthy cells,x3the density of short-term stem-like leukemic cells ,x4 the density of mature leukemic cells.
The time necessary for a ST-HSC to complete a cycle of self-renewal, asym- metric division or differentiation isτ1lfor leukemic cells and τ1h for the healthy ones, while the time necessary for the maturation of leukocytes is denoted byτ2l
in the case of leukemic cells andτ2h for the healthy ones.
As we mentioned in the introduction, experimental evidences attest that the imatinib therapy affects primarily the proliferation rate and secondary, the apoptosis rate. In view of this fact, we consider the treatment functions fu =
1
1−u and f1a = (γ1h−γ1l)u1, f2a = cγ2hu2, with u, u1, u2 : [0, T] → [0,1], whereu(t), u1(t), u2(t) are the treatment effects. The action of treatment on the proliferation rate will be considered throughfu in the function of self-renew βl
and in the function of differentiation or asymmetric division kl. Note that, in this way, both βl and kl became decreasing functions of u. If no drug is given (i.e.u(t) = 0) thenβl((x1+y1)fu) =βl(x1+y1), kl((x1+y1)fu) =kl(x2+y2) and also, a maximal effect happens for u(t) = 1 when the process of division essentially stops (βl≡kl≡0).
Treatment will be consider to act only on the leukemic stem cells compart- ment. The treatment acts on the apoptosis of mature CML cells through f2a
and on the apoptosis of stem-like CML cells throughf1a, restoring this rate to a value closed to the mortality rate of healthy cells. From the law of the mass, we have ˜f1a=Rt
t−τ1lu1(s)ds.. The optimal control model is
˙
x1=f1(x1, x2, y1, y2, x1τ1h, x2τ1h, y1τ1h, y2τ1h)
˙
x2=f2(x2, x1τ2h, x2τ2h, y2τ2h)
˙
y1 =f3(t, x1, x2, y1, y2, x1τ1l, x2τ1l, y1τ1l, y2τ1l, u1, u, uτ1l)
˙
y2 =f4(y2, x2τ2l, y1τ2l, y2τ2l, u2, uτ2l)
(1)
where
f1=−γ1hx1−(η1h+η2h)kh(x2+y2)x1−(1−η1h−η2h)βh(x1+y1)x1+ +2e−b1hτ1h(1−η1h−η2h)βh(x1τ1h+y1τ1h)x1τ1h+
+η1he−b1hτ1hkh(x2τ1h+y2τ1h)x1τ1h f2=−γ2hx2+Ahkh(x2τ2h+y2τ2h)x1τ2h
f3=−(γ1l+f1a)y1−[(η1l+η2l)kl((x2+y2)fu) + (1−η1l−η2l)βl((x1+y1)fu)]y1+ +[2e−b1lτ1l−˜f1a(1−η1l−η2l)βl((x1τ1l+y1τ1l)fuτ1l)+
+η1le−b1lτ1l−˜f1akl((x2τ1l+y2τ1l)fuτ1l)]y1τ1l
f4=−(γ2l+f2a)y2+Alkl((x2τ2l+y2τ2l)fuτ
2l)y1τ2l subject to minimization of the cost functional
minJ(u), (2)
where
J(u) =ay1(T) +by2(T) + Z T
0
u2(t) +u21(t) +u22(t) dt
withg(y(T)) =ay1(T) +by2(T) the weighted sum of the final tumor population andL(u(t)) =
T
R
0
u2(t) +u21(t) +u22(t)
dt, the cumulative drug toxicity.
In this paper we denote Xτ = X(t−τ), where X = (x1, x2, y1, y2). The history of the state variables is given by
X(θ) =ϕ(θ), θ∈[−τmax,0], τmax= max(τ1h, τ2h, τ1l, τ2l).
It is not difficult to see that if the initial conditions have all components positive functions, the solutions of the system will have positive components on all the interval of existence. Indeed, if x1(θ)>0 for any θ ∈ [−τ,0] and there exists T >0 such thatx1(T) = 0 the derivative ˙x1(T) will be positive and this leads to a contradiction. The same argument works for the other components, too.
We assume that: a percentage η1α, α = h, l of stem-like cells population is supposed to undergo asymmetric division; a percentage η2α, α=h, l of the population differentiate symmetrically and the percentage (1−η1α−η2α), α=h, l of the population is supposed to self-renew (see also [13]).
Furthermore,it is assumed that homeostatic mechanisms mantain the hematopoi- etic stem cell population at a constant level. In this respect, the rate of self- renewal is given by a Hill function
βα(X) =β0α θm1
θm1 +Xm, α=h, l
and the rate of differentiation, through symmetric or asymmetric division is supposed to be dictated, through a feedback law, by
kα(X) =k0α θ2n
θ2n+Xn, α=h, l.
Because in this paper we consider competition between healthy and CML cell populations, both this rates will depend on the sum of stem-like respectively mature populations (similar approaches on competition were modeled in [23, 22]).
Forα=h, l,the other parameter are defined as follows:b1αaccounts for the death rate of stem cells and a positiveKαfor the loss rate due to differentiation into other cell lines - the resulting loss rate is denoted asγ1α =Kα+b1α; β0α
and k0α represent the maximal rate of self-renewal, respectively of asymmetric division or differentiation into leukocyte line;θi, i= 1,2, is the value for which βα, respectivelykα attains half of their maximum value;γ2α is the mortality of mature cells;Aαis an amplification factor of mature cells due to differentiation;
mis the parameter controlling the sensitivity of the mitotic re-entry rate βα to changes in the size ofG0andnis the parameter controlling the sensitivity of the
A control DDE model of normal and leukemic cells under treatment 259
asymmetric division or differentiation ratekα to changes in the size of mature population.
Existence of an optimal control The existence of an optimal control results from transforming the problem into an optimal control problem for a system of ordinary differential equations (see next section) whose solutions will be bounded together with their derivatives on compact intervals (see [5]).
3 Discretization of the optimal control problem
In this section, we apply the numerical procedure from Gollmann et al. [10], in order to solve the delay optimal control problem (1)+(2) (see also [8, 9]). For that matter, we write the cost functional in the Mayer form
J(u, y) =h(y(T)), y= (y1, y2)∈R2.
In our case, the reduction of the more general cost functional (2) to Mayer form, proceeds by the introduction of the additional state variablezthrough the delayed equation
˙
z(t) =L(u(t)) so ˙z(t) =u2(t) +u21(t) +u22(t), z(0) = 0.
Then, the cost functional (2) is rewritten as
J(u, y, z) =g(y(T)) +z(T).
In the following, letτ >0 such thatτ1h =k1τ, τ2h =k2τ, τ1l =k3τ, τ2l=k4τ, ki∈N∗, i= 1,4,T =N τ and use the Euler integration method with a uniform step size τ > 0. Of course,τ can be refined in order to obtain an appropriate smaller step-size.
Using the grid pointsti =iτ,i= 0, N and the approximationsx1(ti)'x1i∈ R,x2(ti)'x2i ∈R,y1(ti)'y1i∈R, y2(ti)'y2i∈R, u(ti)'ui, u1(ti)'u1i andu2(ti)'u2i, the treatment functionf1a becomesPk3
j=1u1i−jτand the delay control problem (1)+(2) is transformed into the nonlinear programming problem (NLP)
M inimize J =g(xN, yN) +zN (3)
subject to
x1i−x1i+1+τ f1(x1i, x2i, y1i, y2i, x1i−k1, x2i−k1, y1i−k1, y2i−k1) = 0 x2i−x2i+1+τ f2(x2i, x1i−k2, x2i−k2, y2i−k2) = 0
y1i−y1i+1+τ f3(x1i, x2i, y1i, y2i, x1i−k3, x2i−k3, y1i−k3, y2i−k3, ui, u1i, u1i−1, .., u1i−k3, ui−k3) = 0 y2i−y2i+1+τ f4(y2i, x2i−k4, y1i−k4, y2i−k4, u2i, ui−k4) = 0
zi−zi+1+τ u2i +u21i+u22i
= 0
(4)
−ui ≤0, ui−1≤0, (5)
−u1i ≤0, u1i−1≤0,
−u2i ≤0, u2i−1≤0.
i= 0, N−1.
Herein, the initial value profileϕ1,ϕ2, ϕ3 andϕ4 gives the values x1−i : =ϕ1(−iτ), i= 0, k1
x2−i : =ϕ2(−iτ), i= 0, k2
y1−j : =ϕ3(−jτ), j= 0, k3
y2−j : =ϕ4(−jτ), j= 0, k4
The variable to be optimized is represented by the vector
w= u0, u10, u20, x11, x21, y11, y21, z1, ..., uN−1, u1N−1, u2N−1, x1N, x2N, y1N, y2N, zN
∈R8N. The numerical procedure described above is applied in the next section and
the graphs for states and control for different sets of parameters are obtained.
4 Numerical results and simulations
In the following figures, we plotted the trajectories of the healthy, respectively CML cell populations for the competition system, showing a comparison between the dynamics of a system without treatment and the dynamics of a system subject to optimal control of treatment. To solve the problem of optimal control the Matlab solver for NLP problems fmincon was used, selecting the ’interior- point’ solver.
In all figures, for the healthy cell populations, we choose the same set of pa- rameters value:η1h= 0.7, η2h = 0.1, τ1h= 2, τ2h= 4, γ1h = 0.1, γ2h= 2.4, Ah= 922,β0h = 1.77, k0h= 0.1 , θ1h = 0.5·106, θ2h = 0.36·108. For leukemic cell populations, we consider alteration of the value of the following parameters:
−smaller percent of asymmetric division (η1l);
−bigger percent of self-renewal (1−η1l−η2l);
−lower rate of apoptosis of leukemic stem cells (γ1l);
−lower rate of apoptosis of leukemic mature cells(γ2l);
−enhanced differentiation (Al).
In the following example (Figures 1, 2) all these features were modified.
If we maintain the configuration of parameters from the previous example but consider that the percentage of self renewal of leukemic cells is the same as the percentage of self renewal of healthy cells, we see another manifestation of the disease, for which the treatment dose is lower than in the previous case (Figures 3, 4).
Discussion
The plots of optimal controls (Figures 2 and 4) exhibit an optimal control effect almost constant at 0.5 respectively at 0.25 until the 90th day for all controls.
One can intuitively expect that the treatment effect is proportional with the
A control DDE model of normal and leukemic cells under treatment 261
0 50 100
0 0.05 0.1 0.15 0.2 0.25 0.3
t x1(t)
0 50 100
0 2 4 6 8 10
t x2(t)
States: T=100
1.Healthy cells parameters:η1h=0.70,η2h=0.10,τ1h=2.00,τ2h=4.00,γ1h=0.10,γ1h=2.40, k0h=0.10,Ah=922
2.Leukemic cells parameters:η1l=0.10,η2l=0.50,τ1l=2.00,τ2l=4.00,γ1l=0.03,γ2l=0.80, k0l=0.10,Al=1843
0 50 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t y1(t)
0 50 100
0 20 40 60 80 100 120 140 160
t y2(t)
without treatment with optimal control
Fig. 1. Comparison between the dynamics of a system without treatment and with with optimal control of treatment. The results correspond to the best local minimal solution obtained for different initial guesses ( the value of cost functinal was improved from 1597 to 901)
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t
u(t)
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t
u1(t)
Control: T=100
1.Healthy cells parameters: η1h=0.70,η2h=0.10,τ1h=2.00,τ2h=4.00,γ1h=0.10,γ2h=2.40, k0h=0.10,Ah=922
2.Leukemic cells parameters: η1l=0.10,η2l=0.50,τ1l=2.00,τ2l=4.00,γ1l=0.03,γ2l=0.80, k0l=0.10,Al=1843
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t
u2(t)
Fig. 2.Optimal control of treatment. The controls u, u1, u2 represent the influence of drug on the proliferation rate and apoptosis. One can observe that the drug influence is almost constant and the evolution of u, u1 and u2 are similar if parameters a an b of cost functional are 1.
0 50 100 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
t x1(t)
0 50 100
0 2 4 6 8 10 12 14
t x2(t)
States: T=100
1.Healthy cells parameters:η1h=0.70,η2h=0.10,τ1h=2.00,τ2h=4.00,γ1h=0.10,γ1h=2.40, k0h=0.10,Ah=922
2.Leukemic cells parameters:η1l=0.40,η2l=0.40,τ1l=2.00,τ2l=4.00,γ1l=0.03,γ2l=0.80, k0l=0.10,Al=1843
0 50 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t y1(t)
0 50 100
0 5 10 15 20 25 30 35 40 45 50
t y2(t)
without treatment with optimal control
Fig. 3. Comparison between the dynamics of a system without treatment and with with optimal control of treatment. The results correspond to the best local minimal solution obtained for different initial guesses ( the value of cost functinal was improved from 641 to 260)
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t
u(t)
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t
u1(t)
Control: T=100
1.Healthy cells parameters: η1h=0.70,η2h=0.10,τ1h=2.00,τ2h=4.00,γ1h=0.10,γ2h=2.40, k0h=0.10,Ah=922
2.Leukemic cells parameters: η1l=0.40,η2l=0.40,τ1l=2.00,τ2l=4.00,γ1l=0.03,γ2l=0.80, k0l=0.10,Al=1843
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t
u2(t)
Fig. 4.Optimal control of treatment. The controls u, u1, u2 represent the influence of drug on the proliferation rate and apoptosis. One can observe that the drug influence is almost constant and the evolution of u, u1 and u2 are similar.
A control DDE model of normal and leukemic cells under treatment 263
cell concentration of imatinib; however, in order to study the influence of the prescribed drug concentration on the population’s time-evolution, imatinib phar- macokinetics (PK) and pharmacodynamics (PD) need to be taken into account (see also [20]). Nevertheless, for an optimal effect of treatment, the prescribed dose should be adapted in view of the disease parameters of a certain patient.
In that respect, a competition model of healthy vs. CML cell population that takes into account the influence of PK and PD of imatinib, is subject of further research.
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